Multiply the following complex numbers, marked as blue dots on the graph: $(4 e^{23\pi i / 12}) \cdot (2 e^{13\pi i / 12})$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $4 e^{23\pi i / 12}$ ) has angle $\frac{23}{12}\pi$ and radius $4$ The second number ( $2 e^{13\pi i / 12}$ ) has angle $\frac{13}{12}\pi$ and radius $2$ The radius of the result will be $4 \cdot 2$ , which is $8$ The sum of the angles is $\frac{23}{12}\pi + \frac{13}{12}\pi = 3\pi$ The angle $3\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $3\pi - 2 \pi = \pi$ The radius of the result is $8$ and the angle of the result is $\pi$.